IFAC Conferene on Advane in PID Control PID'1 Breia (Italy), Marh 8-3, 1 Automati deign of robut PID ontroller baed on QFT peifiation R. Comaòliva*, T. Eobet* J. Quevedo* * Advaned Control Sytem (SAC), Tehnial Univerity of Catalonia (UPC) Rambla Sant Nebridi, 1, 8 Terraa (Spain) (Email: [ramon.omaoliva, terea.eobet, joeba.quevedo]@up.edu ) Abtrat. Thi paper propoe an optimization algorithm for the automati deign of robut PID ontroller uing Quantitative Feedbak Theory (QFT) peifiation. The propoed algorithm i baed on a riterion to minimize the energy of the ontrol effort. To illutrate the methodology, the pith angle of a laboratory heliopter i ued a a model appliation with trutured unertainty. The reult how that the deign of robut ontroller an be formulated uing an objetive funtion and a number of retrition that are developed a peifiation. Keyword: Robut Control, PID ontroller, Quantitative Feedbak Theory, Automati Loop Shaping, Heliopter 1. INTRODUCTION Thi artile propoe a method for the automati deign of a robut proportional integral derivative (PID) ontroller that ombine the Quantitative Feedbak Theory (QFT) QFT automati loop haping method with optimization tehnique to automatially deign PID ontroller minimizing the ontrol effort. The deign method applie to the ae tudy whih onit of the pith angle ontrol of a laboratory heliopter with trutured unertainty (Garia- Sanz et al., 6). The QFT tehnique an be ued to deign ontroller with no predetermined truture for ytem with parametri unertaintie that are modeled by intervalar linear model. The lai deign of ontroller to enure that peifiation are met involve the frequeny repone, where the ontroller gain i attenuated by appropriately manipulating the pole and zero of the tranfer funtion of the ontroller. Thi deign an be implemented quite effiiently uing upport oftware, uh a the Matlab QFT toolbox (Borgheani et al., 1994). If a ontroller with a fixed low order truture i uppoed, the deigner ha to manipulate the parameter of the ontroller to enure that all peifiation of the loed loop are met. An open problem in QFT i the automati deign of ontroller, whih i known a automati loop haping (ALS). The idea of the automati deign of ontroller QFT wa introdued by Gera and Horowitz (199). In the literature, there are variou approahe to the automati deign of loop ontroller for a plant with unertainty, o that peifiation lead to robutne (tability, et.) and minimize the ontroller gain. Tehnique that have been ued inlude geneti algorithm (Chen et al., 1998), linear programming (Nataraj and Nandkihor, 7) and optimization algorithm (Chait et al., 1997), (Sahin, 5). Some author (Nandakumar et al., ) ued a hybrid tehnique involving optimization and propagation of unertainty, ombining interval global optimization and loal nonlinear optimization. It i uual to propoe ALS method fixing the ontroller truture, for example in Yaniv and Nagurka (1995). Nataraj and Dehpande (8) propoe the ynthei of a ontroller a a ontraint atifation problem with interval variable. Zolota and Halikia (1999) propoe an ALS method for PID ontroller baed on earhing over a dene et of ontroller. Molin and Garia-Sanz (9) deribe a QFT method for deigning ontroller that i baed on a ombination of tehnique, evolutionary algorithm and geneti algorithm. The ame author, Garia-Sanz and Molin (1) propoed an ALS method, by grouping previouly all uually QFT peifiation into two expreion, one in term of enitivity funtion, and other in term of the omplementary enitivity funtion. In thi paper, a loop-haping deign for PID ontroller that minimize the energy of the ontrol effort by mean of optimization tool i propoed. A the problem to olve i non linear and non onvex, the ommerial global optimization pakage Tomlab (Pintér et al., 5), whih i baed on branh and bound algorithm, i ued. In Setion we formulate an automati ontroller deign problem, to minimize the energy of the ontrol effort. Setion 3 how the reult of the deign applied to a laboratory heliopter model ytem, and Setion 4 preent the main onluion.. DESIGN METHOD FOR ROBUST CONTROLLERS.1 Formalizing the problem The uual ontrol truture in the QFT tehnique i hown in Figure 1, where P, G, F and H repreent the tranfer funtion
IFAC Conferene on Advane in PID Control PID'1 Breia (Italy), Marh 8-3, 1 of the plant, the ontroller, the pre-filter and the enor, repetively. R, Y, U, E and Q repreent the et point, the output, the ation ontrol, the ontrol error and the enor output ignal. Finally, W, V, D and N indiate added input diturbane at the input of the ontroller, the input of the plant and the output of the plant a well a noie diturbane, repetively. Aording to Bhattaharyya et al. (1995), an unertain plant an be deribed by an interval model: P, B, (1) A, where B and A are polynomial in the domain, and i the vetor of unertain parameter of dimenion p with their value bounded by a ompat et of box type, i.e. p Seondly, the requirement for diturbane rejetion at the plant output, or enitivity, i expreed a: 1 S, j 1 L, j where ( ) i the magnitude of the diturbane rejetion, that i the upper bound of the magnitude of the tranfer funtion. Finally, the ontrol effort peifiation i expreed a: U,, Y (4) j D j T j G j S j (5) where S, j 1 1 L, j i the enitivity, and i the magnitude of the ontrol effort, that i the upper bound of the frequeny magnitude of the tranfer funtion U j W j.. Propoed deign methodology Fig. 1. QFT generi ontrol truture A laial problem in QFT (Horowitz, 1993; Houpi, and Ramuen, 1999; Yaniv, 1999) i to yntheize the feedbak ontroller G() and the prefilter F() a tritly proper, rational and table tranfer funtion. Thu, that ome peifiation are atified, while the bandwidth of the ontroller i kept a low a poible, depite the preene of unertainty in P(). In general, feedbak ontrol i defined a G k, B k, () A k, where B and A are polynomial in the domain; and k i the vetor ontroller parameter of dimenion g. Controller deign uing QFT i arried out by formulating everal of the ytem frequeny peifiation, aording to the requirement. In thi tudy, three type of peifiation are ued: robut tability, robut diturbane attenuation, and minimization of the ontrol effort. When the nominal open loop i given a L, j P, j G j and H()=1 and F()=1, a robut tability peifiation i expreed a: T R L L, j, j 1( ) 1, j where 1 ( ) i the maximum magnitude of the loed-loop traking tranfer funtion Y j R j. (3) In thi paper, we propoe an algorithm for automati loop haping. It i baed on minimizing the ontrol effort impule repone energy, given by (5), for a plant with a partiular value of θ and ontroller k : U E(, k ) T (,, k ) d (6) while the energy for the whole family of plant will be given by E(, k ) d T (,, k ) dd (7) U A direte approximation of (7), given a direte grid of frequenie 1,,, nw and a grid of unertain parameter of the plant 1,,, np, ould be: np nw E(, k ) d T (,, k ) (8) U i j i1 j1 where np and nw are the number of plant and frequenie onidered, repetively. The Algorithm how the proedure to ompute the optimal olution of the ontroller parameter k KC, where KC i the initial bounded et of ontroller parameter, g KC k k k k, whih minimize the ontrol effort (8) taking into aount the retrition gave by (3), (4) and (5), in ae of a generi ontroller truture. Moreover, although it i not a predefined requirement, we hoe a truture of ontrol that i imilar to the one ued in
IFAC Conferene on Advane in PID Control PID'1 Breia (Italy), Marh 8-3, 1 the referened artile, i.e. a PID with a filter in the derivative ation: 1 T d Gk, K p(1 ) T T 1 where K p i the proportional gain, T i i the integral time, T d i the derivative time and T f i the high frequeny filter parameter. Uually filter i not been regarded a a part of the deign but added afterward to prevent that the high frequeny gain of the ontroller growing up. In the propoed approah the four parameter ha been onidering, being the ontroller parameter vetor k K p, Ti, Td, T. In thi f ae, the et of a minimum and maximum parameter bound are k K p, Ti, Td, T f, k K p, Ti, Td, T f. Algorithm. Automati loop haping Step 1. Chooe the grid of frequenie 1,,, nw and parameter of the plant i 1,,, np i. Step. Solve the following optimization problem. For eah frequeny 1,,, nw, and for the et of plant parameter : Subjet to: min T (,, k ) k For i=1, np End np nw i1 j1 For j=1, nw End T S T k k k U i j k 1,, ( ) R i j j i, j, k j,, k U i j j f Step 3. If the olution ha not been found, the ontroller parameter bound hould be inreaed or/and the peifiation hould be relaxed and go to Step 1. Fig.. Algorithm for the automati deign of the ontroller Wallén et al. () preent a tudy of the advantage to ontrain the ratio T i /T d to a fixed value n4 when loop haping i applied to deign PID ontroller. Alo a low pa high frequeny filter, with a priori eleted time ontant T f, i ued in the derivative term in order to limit the ueptibility of the ontroller to meaurement noie. T f ha been eleted a one tenth of the derivative time following the uggetion of Atrom et al. (1995). (9) Aording thi, we fixed ome retrition to obtain onitent parameter, and to redue the earhing pae of PID parameter; ome retrition have been fixed between them: T T / 4; T.1* T (1) d i f d Thee two ondition are added in the algorithm a new ontraint. In ummary, the PID automati loop haping i baed to olve the optimization problem (Algorithm, Fig. ), adding the new mentioned ontraint (1), for all i and i. 3. APPLICATION TO THE MODEL OF A LABORATORY HELICOPTER 3.1 Phyial model of the heliopter To tet the Algorithm propoed in the previou etion, we ued a ale laboratory heliopter with two parallel rotor. Thi appliation wa deigned by Quaner and deribed in detail in Garía-Sanz et al. (6). Fig. 3 how it general appearane. Fig. 3. General appearane of the Quaner Heliopter 3 DOF Garía-Sanz et al. (6) preent a detailed deription of the heliopter harateriti, and the phyi equation that an be ued to obtain a linear time invariant model (LTI) of the pith angle, by preproeing appropriate linearization and implifiation. The final model provided a eond order LTI model, with parametri or trutured unertainty obtained from experimental data. A more omplete model i deribed in Egaña et al. (1) and Garía-Sanz et al. (), in whih the angle of roll, pith and yaw are inluded a three degree of freedom (3 DOF). The pith angle model i ued to how the reult of the Algorithm deribed in Fig.. From the etimate of phyial parameter (length, ma, frition) in Egaña et al. (1), a eond order tranfer funtion i defined in whih the ytem ignal input i voltage, the output i the pith angle and the parameter are unknown but bounded, a it i indiated in (11).
Magnitude (db) Open-Loop Gain (db) IFAC Conferene on Advane in PID Control PID'1 Breia (Italy), Marh 8-3, 1 P kn n n 3. Deign peifiation k.7,.1.1,.16 n.55,.6 (11) The aim of the robut ontroller deign are imilar to thoe propoed in Egaña et al. (1), partiularly robut tability, enitivity and ontrol effort. Regarding thi previou deign, the following limit of performane have been eleted: 1 ( ), ( j.1)( j.3) (( j) 4 j4) and 5. Firt and third, ontrain the ratio between output-input and ontrol effort-input, repetively, at all range of frequenie. Seond, enitivity bound i a frequeny dependent peifiation. In the frequeny petrum, the ame range wa taken for all the peifiation and only for low frequenie:.1, 1 rad/e. We hoe thi frequeny range beaue it inlude the frequenie that are apparently more problemati in term of meeting the peifiation. The domain of the ontrol parameter ha been eleted taking into aount the previou work of Egaña (1) and initial reult of our methodology with a large domain of ontrol parameter. Now, we retrit thi domain to the following bounded et (time parameter in eond): K p 1, 3 Ti 1, Td.5, 5 Tf.5,.5 (1) illutrate the method. When we inreaed the number of plant and the frequeny grid, we found that the method wa till valid, but the tradeoff wa a higher omputational ot. By mentioned ondition, the optimal PID ontroller omputed i: 93.76(1.5434.4941) G (.591 1) (13) orreponding to a ontroller (9) with the following parameter value (time parameter in eond): K.; T.37; T.59; T.59; (14) p i d f 4 3 1-1 - -3-4.1 1.5 5 1 5 1-36 -315-7 -5-18 -135-9 -45 Open-Loop Phae (deg) Fig. 3. Nihol hart with the deigned ontroller. That inlude the three peifiation - Bode enitivity peifiation 3.3 Reult -4 In order to olve the optimization of the Algorithm that orrepond to a non onvex and non-mooth problem, a global optimization ommerial toolbox ha been ued to olve thi problem. In partiular, the optimization environment Matlab/Tomlab toolbox (Pintér et al., 5) uing the Effiient Global Optimization algorithm, ego, (Jone et al., 1998) ha been applied. To obtain the optimal ontroller olution, objetive funtion (8) and ontraint (3), (4), (5) and (1), ha been properly manipulated to obtain the equivalent polynomial expreion. The Algorithm, propoed in Setion., wa applied by taking a finite number of 8 plant, obtained by all the ombination of lower and upper parameter bound, and hooing the following grid of frequenie:.1 1.5 5 1 5 1 rad / e. We eleted thi limited number of plant and frequenie beaue we onidered that it provided enough information to -6-8 -1-1 1-4 1-3 1-1 -1 1 1 1 1 Frequeny (rad/e) Fig. 4. Senitivity peifiation and Bode diagram module of a et of 8 ontrolled plant. The QFT toolbox (Borgheani et al., 1994) allow validate the PID omputed. Fig. 3 how, in the Nihol hart, the QFT bound for eah i. Alo, in a olid blak line, how the nominal open loop plant with the yntheized ontroller, L(j i ) (the irle o denote the repone at eah i frequeny), where nominal plant orrepond with all the parameter at their lower value. Notie that L(j i ) atifie
IFAC Conferene on Advane in PID Control PID'1 Breia (Italy), Marh 8-3, 1 the bound at ome frequenie at the limit, onluding that the optimal ontroller were obtained, for the et of frequenie hoen. A redundant information, there i hown in Figure 4 the module on the Bode diagram of the enitivity peifiation. The red line how the enitivity module for the eight plant eleted and the green line how the maximum magnitude of the diturbane rejetion, ( ), to note that it i the upper bound of the magnitude of the Y j D j tranfer funtion. the abrupt jump in the graph that appear to be modified. The optimal ontroller orrepond to the lowet point of the boundary line, whih oinide with the value of Kp and Ti found by the optimizer given in (14). The value of the objetive funtion for the optimum point (minimum) i fopt=.54e+7. All the ontraint are atified for thi optimal point, orreponding to the ontroller found. 3.4 Time domain analyi The deribed benhmark (Garía-Sanz et al., 6) propoe the deign of a ontroller that an follow a given temporal referene ignal, a hown in Fig. 7, where are alo hown the time repone of eight plant hoen at random. Fig. 7 how that the repone of the loed loop ytem i fat, with a peak at around % when the input ignal i a tep. Thi peak ould be redued by lowering the threhold value of robut tability. The peed of the repone entail a ignifiant inreae in ontrol effort. Thi oillation an be redued by dereaing the value of the ontrol effort. In thi ae, a large value i onidered a a ontraint. An additional information i available in Fig. 5. The objetive funtion i omputed uing equation (8) for different value of ontroller parameter, Kp [1, 3] and Ti [1, ], where Td and Tf have been retrited by (1). Funtion with ontraint 7 x 1. -. -.4 4 6 Time () 8 1 1 8 1 1 Linear Simulation Reult. -. -.4 9.4 Angle output, 8 plant(rad) Fig. 5. Repreentation of the funtion to minimize, without onidering ontraint and depending on the parameter Kp and Ti Angle referene Set Point(rad).4 4 6 Time (e) 8 7 Fig. 7. Time repone of a et of 8 plant in repone to a referene et point 6 5 4 3 f =.54e+7 opt 4. CONCLUSIONS 1 3 5 15 1 Kp 1 5 15 Ti Fig. 6. Repreentation of the funtion to minimize, onidering ontraint, depending on parameter Kp and Ti When ontraint are added, only one of them, peifially the enitivity peifiation (4), i not fulfilled for the whole range of the parameter Kp and Ti. Thi i hown in Fig. 6 by In thi paper, we propoed an algorithm for the automati deign of ontroller. It i baed on optimization of an index aoiated with ontrol effort. Some tudie in the literature aim to obtain a imilar objetive of automati loop haping baed on optimization. The algorithm ha been applied to the peifi ae of ontroller deign for a model of the pith angle of a laboratory heliopter, with trutured unertainty. It wa upported by a ommerial global optimization tool, Tomlab, with atifatory reult, a validated by imulation.
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